### 46th Putnam 1985

**Problem B4**

Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the x-axis and y-axis, lies entirely inside (or on) C?

**Solution**

Answer: 4/π^{2}.

Given P, let R be the rectangle inscribed in C with sides parallel to the axes. Then the rectangle with diagonal PQ lies entirely inside C iff Q lies inside R.

Let O be the center of C. Let the diameter of C parallel to the y-axis make an angle θ with OP. Then the area of R is 4 sin θ cos θ. So the required probability is 2/π ∫_{0}^{π/2} (4 sin θ cos θ)/π dθ = 4/π^{2} ∫_{0}^{π/2} sin 2θ dθ = 4/π^{2}.

46th Putnam 1985

© John Scholes

jscholes@kalva.demon.co.uk

7 Jan 2001